I have a constrained maximization problem over 2 variables (say, x and y) that also includes some unknown, but fixed, parameters (say, a, b, and c). However, I know some things about a, b, and c (a>1, 0<b<1, 0<c<1).

I would like to tell Mathematica that a, b, and c have these properties so that it will return a maximum that is not a list of rules, most of which are not pertinent since the above inequalities are not satisfied. In addition, I want to pass these rules to mathematica before running the maximization problem because, since the maximization problem I'm running is very complicated when the inequalities on the fixed parameters are not satisfied, the program is taking too long to run.

For example:

Maximize[{a*x^2,-1<=x<=1},x]

will return a list of rules, based on what a is.

However, if we know that a is negative, we know that the maximizer is x=0. If we know that a is positive, we know that the maximizer is x=-1 or x=1.